Abstract: We consider simultaneous inference of the time-varying correlation, as a function of time, between two nonstationary time series, when their trend functions are unknown. Unlike the stationary setting, where the effect of precentering using the sample mean is trivially negligible, in the nonstationary setting, it is difficult to quantify the effect of precentering using nonparametric trend function estimators. This is mainly because the trend estimators are time-varying across different time points, which makes it difficult to quantify their cumulative interaction with the error process in a time series setting. We propose using a centering scheme that, instead of aligning with the time point at which the data are observed, aligns with the time point at which the local correlation estimation is performed. We show that the proposed centering scheme leads to simultaneous confidence bands with a solid theoretical guarantee for the time-varying correlation between two nonstationary time series when their trend functions are unknown. Lastly, we demonstrate the proposed method using numerical examples, including a real-data analysis.

Key words and phrases: Kernel smoothing, local linear estimation, noncentered data, simultaneous confidence band.